Question: Multiply and simplify the following complex numbers: $({2+5i}) \cdot ({5-3i})$
Solution: Complex numbers are multiplied like any two binomials. First use the distributive property: $ ({2+5i}) \cdot ({5-3i}) = $ $ ({2} \cdot {5}) + ({2} \cdot {-3i}) + ({5i} \cdot {5}) + ({5i} \cdot {-3i}) $ Then simplify the terms: $ (10) + (-6i) + (25i) + (-15i^2) $ Imaginary unit multiples can be grouped together. $ 10 + (-6 + 25)i - 15 i^2 $ After we plug in $i^2 = -1$, the result becomes $ 10 + (-6 + 25)i - (-15) $ The result is simplified: $ (10 + 15) + (19i) = 25+19i $